well, first thing i urge you not to go the 5.1 surround route. you really can't do that anyways without getting a dedicated center chan speaker and amp. you can use the pac adapter (or the soundgate adapter which is supposed to be of higher quality) to retain your stock woofers. that will allow you to retain the stock sub but you will need a small amp to power the other speakers.
you could use the stock wiring but since the amp is in the back you're better off making new runs to the speakers. the corvette is the easiest car to run wiring in, i doubt it'll take you much longer to run wire to the doors then it would running wires from the amp to the harness behind the radio.
my personal recommendation would be to dump the idea of keeping the stock speakers and the 5.1 setup and drop the rear speakers as well.
since you were planning on running the 4 stock speakers you were going to need a four chan amp and a bose interface. Stick with the 4 chan amp, now the bose interface is about $100. if you put that money towards raw drivers such as:
which are on sale right now for $31 each
and then buy some seas neo tweeters (or northcreek tweeters) for about the same. you are going to achieve far far superior sound for about $25 difference.
here is some ref: